Visualizing vectors in 2 dimensions | Two-dimensional motion | Physics | Khan Academy

Main ideas and lessons

• Moving beyond one dimension: Earlier problems often treated motion as 1D (forward/back, right/left, or up/down). This video extends the idea to 2D (and notes the generalization to 3+ dimensions).

• What a vector is: A vector has:

  • Magnitude (the arrow’s length)
  • Direction (the direction the arrow points)

• How to add vectors visually in 2D:

  • Vectors can be shifted (moved) without changing them, as long as they keep the same magnitude and direction.
  • To compute A + B:
    1. Draw A
    2. Shift B so its tail starts at the head of A
    3. The vector from the tail of A to the head of the shifted B is C = A + B
  • Interpretation example (displacement): If you displace by A and then by B, the net displacement is the vector sum.

• Decomposing a 2D vector into components:

  • Any 2D vector X can be written as the sum of its components:
    • X = X_horizontal + X_vertical
  • This turns a 2D problem into two separate 1D problems:
    • one along the horizontal axis
    • one along the vertical axis

• Using trigonometry to find component magnitudes (math method):

  • Define vector A with magnitude 5, making an angle 36.8699° with the positive x-axis.
  • Components:
    • A_y (vertical component; “opposite” the angle)
    • A_x (horizontal component; “adjacent” to the angle)
  • The components form a right triangle:
    • hypotenuse = |A| = 5
    • opposite side = |A_y|
    • adjacent side = |A_x|

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Detailed instruction-style bullet points (methodology)

1) Vector addition in 2D (graphical method)

• Draw vector A with the correct length and direction.
• Draw vector B with the correct length and direction.
• Shift vector B so that:
  • the tail of B is placed at the head of A
• The resulting sum C = A + B is:
  • the vector that starts at the tail of A and ends at the head of the shifted B

2) Decompose a vector into horizontal/vertical components (conceptual)

• For a vector X in 2D:
  • construct a horizontal component and a vertical component that add tip-to-tail to X
• Represent it as:
  • X = (horizontal component) + (vertical component)
• Use this to split a 2D problem into:
  • a horizontal 1D problem
  • a vertical 1D problem

3) Compute component magnitudes using trig (given magnitude and angle)

Given:

• Vector A magnitude = 5
• Angle θ = 36.8699° measured from the positive x-axis

Triangle relationships:

• sin(θ) = opposite / hypotenuse
• cos(θ) = adjacent / hypotenuse

Compute components:

• Vertical component magnitude:
  • |A_y| = |A| · sin(θ) = 5 · sin(36.8699°) ≈ 3
• Horizontal component magnitude:
  • |A_x| = |A| · cos(θ) = 5 · cos(36.8699°) ≈ 4

Result:

• A_y ≈ 3
• A_x ≈ 4
• This matches a 3-4-5 right triangle

4) Why components matter (preview of next idea)

• If a vector represents velocity (example preview):
  • A velocity of magnitude 5 m/s in the given direction can be split into:
    • upward component: 3 m/s
    • rightward component: 4 m/s
• This converts a 2D motion problem into two independent 1D component problems

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Speakers / sources featured

• Khan Academy (Voiceover)